Elliptic operators with unbounded drift coefficients and Neumann boundary condition. (Q1428434)

The authors study the realization \(A_N\) of the linear elliptic operator \[ {\mathcal A}=\tfrac12 \Delta -\langle DU,D\cdot\rangle \] in \(L^2(\Omega,\mu)\) with Neumann boundary condition. Here \(\Omega\) is a possibly unbounded convex open set in \({\mathbb R}^N,\) \(U\) is a convex unbounded function with \(DU(x)\) being the element with minimal norm in the subdifferential of \(U\) at \(x\) and \(\mu(dx)=c \exp(-2U(x))dx\) is a probability measure, infinitesimally invariant for \({\mathcal A}.\) The main result is that \[ A_N\colon\quad \left\{u\in H^2(\Omega,\mu)\colon\;\partial u/\partial n=0,\;{\mathcal A}u \in L^2(\Omega,\mu)\right\} \] is a self-adjoint dissipative operator. The technique of proofs involves a penalization method applied to the family of operators \[ {\mathcal A}_\varepsilon u(x)={1\over 2}\Delta u(x) -\langle DU_\varepsilon(x),Du(x)\rangle,\quad x\in {\mathbb R}^N \] with \[ U_\varepsilon(x)=U(x)+{1\over{2\varepsilon}}(\text{ dist\,}(x,\Omega))^2. \] An additional assumption on the convexity of \(U-\omega| x| ^2/2\) for some \(\omega>0\) allows the authors to show that the measure \(\mu\) satisfies Poincaré and \(\log\)-Sobolev inequalities, whence smoothing properties and asymptotic behavior of the semigroup generated by \(A_N\) follow.